Tail fields generated by symbol counts in measure-preserving systems

Abstract

A finite-state stationary process is called (one or twosided) super-K if its (one or two-sided) super-tail field—generated by keeping track of (initial or central) symbol counts as well as of arbitrarily remote names—is trivial. We prove that for every process (α, T ) which has a direct Bernoulli factor there is a generating partition β whose one-sided super-tail equals the usual one-sided tail of β. Consequently every K process with a direct Bernoulli factor has a one-sided super-K generator. (This partially answers a question of Petersen and Schmidt).